{
  "id": "oai:papers.qnfo.org:stress-induced-switching-from-fibonacci-to-lucas-phyllotaxis-a-quantitative-thre",
  "slug": "stress-induced-switching-from-fibonacci-to-lucas-phyllotaxis-a-quantitative-thre",
  "title": "Stress-Induced Switching from Fibonacci to Lucas Phyllotaxis: A Quantitative Threshold Model",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Plants that develop spiral leaf or floret arrangements predominantly exhibit Fibonacci divergence angles near 137.5^∘, yet agronomic guidelines treat phyllotaxis as a static trait, so growers and breeders cannot anticipate the pattern changes observed under water or nutrient stress. Existing analytical models fix the golden angle and provide no criterion for when a stand will depart from it. We propose a stress-dependent threshold model in which a normalized stress parameter σ interpolates between a Fibonacci-weighted and a Lucas-weighted contribution to the primordium-packing problem, drawing on algebraic identities that link the two sequences. Equating the weighted contributions at sequence index n=5 yields a critical stress σ_c = F₅/(F₅+L₅) = 5/16 = 0.3125; recomputing at n=6 shifts the threshold by only 1.54\\%. The model predicts an abrupt switch of the divergence angle from θ_F = 360^∘/φ² ≈ 137.508^∘ to a Lucas-related angle θ_L = 360^∘/(φ+2) ≈ 99.506^∘, a relative angular loss R = (θ_L-θ_F)/θ_F ≈ -0.2764, that is, a 27.6\\% reduction in divergence angle. If the threshold holds, irrigation and fertilisation schedules could be set to keep a crop below σ_c, preserving golden-angle packing, and the index R gives breeders a single metric for selecting stress-tolerant genotypes.",
  "abstract_markdown": "Plants that develop spiral leaf or floret arrangements predominantly exhibit Fibonacci divergence angles near $137.5^\\circ$, yet agronomic guidelines treat phyllotaxis as a static trait, so growers and breeders cannot anticipate the pattern changes observed under water or nutrient stress. Existing analytical models fix the golden angle and provide no criterion for when a stand will depart from it. We propose a stress-dependent threshold model in which a normalized stress parameter $\\sigma$ interpolates between a Fibonacci-weighted and a Lucas-weighted contribution to the primordium-packing problem, drawing on algebraic identities that link the two sequences. Equating the weighted contributions at sequence index $n=5$ yields a critical stress $\\sigma_c = F_5/(F_5+L_5) = 5/16 = 0.3125$; recomputing at $n=6$ shifts the threshold by only $1.54\\%$. The model predicts an abrupt switch of the divergence angle from $\\theta_F = 360^\\circ/\\varphi^2 \\approx 137.508^\\circ$ to a Lucas-related angle $\\theta_L = 360^\\circ/(\\varphi+2) \\approx 99.506^\\circ$, a relative angular loss $R = (\\theta_L-\\theta_F)/\\theta_F \\approx -0.2764$, that is, a $27.6\\%$ reduction in divergence angle. If the threshold holds, irrigation and fertilisation schedules could be set to keep a crop below $\\sigma_c$, preserving golden-angle packing, and the index $R$ gives breeders a single metric for selecting stress-tolerant genotypes.",
  "date": "2026-11-03",
  "first_published": "2026-11-03",
  "last_revised": "2026-11-03T00:00:00Z",
  "modified": "2026-11-03T00:00:00Z",
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  "doi": null,
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  "url": "https://papers.qnfo.org/papers/stress-induced-switching-from-fibonacci-to-lucas-phyllotaxis-a-quantitative-thre",
  "license_url": "https://legal.qnfo.org/",
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  "version_url": "https://papers.qnfo.org/papers/stress-induced-switching-from-fibonacci-to-lucas-phyllotaxis-a-quantitative-thre/v/1.0.0",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Stress-Induced Switching from Fibonacci to Lucas Phyllotaxis: A Quantitative Threshold Model"
      }
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    "publicationYear": "2026",
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        "description": "Plants that develop spiral leaf or floret arrangements predominantly exhibit Fibonacci divergence angles near 137.5^∘, yet agronomic guidelines treat phyllotaxis as a static trait, so growers and breeders cannot anticipate the pattern changes observed under water or nutrient stress. Existing analytical models fix the golden angle and provide no criterion for when a stand will depart from it. We propose a stress-dependent threshold model in which a normalized stress parameter σ interpolates between a Fibonacci-weighted and a Lucas-weighted contribution to the primordium-packing problem, drawing on algebraic identities that link the two sequences. Equating the weighted contributions at sequence index n=5 yields a critical stress σ_c = F₅/(F₅+L₅) = 5/16 = 0.3125; recomputing at n=6 shifts the threshold by only 1.54\\%. The model predicts an abrupt switch of the divergence angle from θ_F = 360^∘/φ² ≈ 137.508^∘ to a Lucas-related angle θ_L = 360^∘/(φ+2) ≈ 99.506^∘, a relative angular loss R = (θ_L-θ_F)/θ_F ≈ -0.2764, that is, a 27.6\\% reduction in divergence angle. If the threshold holds, irrigation and fertilisation schedules could be set to keep a crop below σ_c, preserving golden-angle packing, and the index R gives breeders a single metric for selecting stress-tolerant genotypes.",
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